In the one-liquid case, thanks to the invariance of the problem in the \(\hat{z}\) -direction, imposing the value of the curvature in the limit \(\hat{z} \rightarrow -\infty \)  is enough to perform the asymptotic matching [1]}.
In our two-liquid case, however, we benefit from the invariance along \(\hat{z}\)  only for one of the interfaces because the relative position of interfaces (I) and (II) far from the plate is fixed (they are separated by a given distance \(\Delta H\) ).
Consequently, the boundary conditions in the limit \(\hat{z} \rightarrow -\infty \)  not only result from imposing the curvatures of the two static menisci, but also require specifying how far apart the interfaces are.
In order to fulfill these two conditions, the film thicknesses \(\hat{h}_i\)  are required to follow the parabolic approximations of the static menisci (Equations (REF ) and ()), expressed in terms of the rescaled variables \(\hat{h}_i = Ca^{-2/3} \, h_i\)  and \(\hat{z} = Ca^{-1/3} \, z\) :
\(\hat{h}_1 & = \frac{1}{2}\sqrt{2\left(1-R\right)} \left(\hat{z} - \hat{z}_{cl,1}\right)^2, \\\hat{h}_2 & = \frac{1}{2}\sqrt{2R/\Sigma }\left(\hat{z} - \hat{z}_{cl,2}\right)^2,\) 
